Over the past year, the majority of my research on this project was performed in two areas: (1) fitting a Hill model to binary dose-response data and (2) generalizing the concept of relative potency. These two areas of research are described in more detail below. Area 1: The Hill model is a popular nonlinear model often used to characterize dose-response data from toxicity studies. When the response data are binary, the Hill model corresponds to a 4-parameter logistic model. Relative to the usual 2-parameter logistic model, the Hill model allows the dose-response curve to have a lower limit that is greater than zero and an upper limit that is less than one. Thus, we can envision three categories of subjects: those who will always respond regardless of dose (which leads to a lower limit above zero), those who will never respond regardless of dose (which leads to an upper limit below one), and those whose risk of response is a function of dose. We can postulate a missing data problem, where subjects are observed to respond or not, but we do not know which responders were "destined" to respond, nor do we know which non-responders were "unsusceptible" to response. We developed an EM algorithm to solve this missing data problem and to obtain the maximum likelihood estimates of the Hill model parameters. The EM algorithm is easy to program, covariates are simple to incorporate, and certain natural constraints are satisfied automatically. In ongoing research, we are studying ways to determine the number of uniquely estimable parameters when a Hill model is fitted to binary data. It is well known that there are identifiability problems if all or none of the subjects respond. Similar problems arise if all subjects receiving a dose below a certain level do not respond and all subjects receiving a dose higher than that same level do respond. On the other hand, if there are several observed response rates and they tend to increase with dose, then typically the Hill model parameters are uniquely estimable. For intermediate cases, however, the number of uniquely estimable parameters appears to be related to the number of distinct nonparametric estimates obtained under a monotonicity constraint (i.e., the number of level sets in an isotonic regression analysis). An article about the EM algorithm has been accepted for publication by the Journal of Agricultural Biological and Environmental Statistics. We are currently preparing a draft manuscript about determining the number of uniquely estimable Hill model parameters. Area 2: Within a class of "similar" chemicals, the relative potency of one chemical compared to another is the ratio of doses producing the same toxicity response, and this ratio does not vary with the level of response. On the other hand, for non-similar chemicals, the relative potency of one chemical compared to another need not be constant and typically varies according to where along the dose-response curves the dose ratio is calculated. We are studying three relative potency functions, which vary with dose level, response level, and quantile of the response range. If two chemicals differ only with respect to their ED50s (i.e., the chemicals are similar), then all three functions are constant and equal to the ratio of the ED50s. Otherwise, the functions are generally not constant, and they may cross one, indicating that one chemical is more potent than the other over some regions (of doses, responses, or response quantiles) and vice versa for other regions. If chemicals are not similar, then inferences based on ratios of ED50s or based on models that force the other parameters to be identical can be misleading. We are currently preparing a draft manuscript about this generalized concept of relative potency. In the near future, we plan to develop formal statistical methods for analyzing dose-response data within this more general context.